Derivatives + Optimization

One of the most interesting and practical applications of derivatives is their ability to look at a relationship or a function, and find a desired outcome, such as maximizing or minimizing a variable. For example, suppose you run a milk business, and you want to decide on the price of a new organic jug of milk. Naturally you would want to select a price that maximizes revenues. So you ask your analyst friend for help to do some analyst stuff. He crunches some numbers and does some studies to find that the price-quantity curve of your new jug of milk can be approximated by the relationship S = 10 – 4lnP, where S represents the sales (in 100,000 units) and P represents the price (we’ll use dollars as our currency to make things simpler). This makes sense because as you can see from the graph, as the price increases, the number of sales decreases.

It turns out your analyst friend went to business school and didn’t have a strong background in Calculus, so he forgot that you wanted to maximize revenues and just gave you a price-quantity function instead. Darn. Well, since we know Calculus, we can find out the answer. Since revenues is just Price * Sales, and the Sales equation is the same as what we showed above, our revenue (in $100,000) can be represented by R=P*(10-4lnP)

This is what the price vs revenue function looks like. Since we have graphing software on our computers, we can see that the optimum price is around $4.50, but we can also find this number analytically by taking the derivative of the price versus revenue function and setting it equal to zero, because the derivative represents the slope of a tangent line. As we can see, whenever a function reaches a maximum or minimum value, the derivative or the slope of a tangent line at that maximum or minimum point would equal zero. So, whenever you want to maximize or minimize something for a desired outcome, whether it be minimizing the risk of an investment, maximizing its returns, maximizing revenue, profit, minimizing cost, or even minimizing the time spent studying for an exam, be sure to think about Calculus for help.